Observable indices of competitiveness of an industry, like the average Lerner index in equa-tion (2.4.1), may be relatively crude measures of true competitiveness. In this secequa-tion we therefore assess the robustness of our estimator, relative to a parametric model estimator like Hashmi’s, to measurement error in the index of competitiveness. We …rst show that both models, as one would expect, become inconsistent if competitiveness is mismeasured, even when the models are otherwise correctly speci…ed. However, we also show that the bias in our estimator resulting from measurement error is quite small relative to alternative estimators.
First consider the case where competitiveness is mismeasured, but a parametric model like Hashmi’s (dropping …xed e¤ects for simplicity) is the correct speci…cation in terms of true competitiveness. This model assumes
ln Y = 0+ 1c + 2c 2+ee; (2.7.1)
where ln Y is logged innovation, c is the true level of competitiveness, andee is an error term.
For simplicity we ignore discreteness in ln Y , and we assume c can be linearly decomposed into the observable instrument V and an unobserved independent component W , so
c = V + W . (2.7.2)
Assume validity of Hashmi’s control function type assumption that ee = W + e where e is independent of W and V , so
ln Y = 0+ 1c + 2c 2+ W + e (2.7.3)
In this model, if c were observed, then control function estimation (…rst regressing c on a constant and V , getting the residuals cW , and then regressing ln Y on a constant, c , c 2, and cW ) would consistently estimate the coe¢ cients and hence any desired treatment e¤ects based on .
Now assume the observable competitiveness measure c equals the true measure c plus measurement error ce, so
c = c + ce; (2.7.4)
where ce is the measurement error and independent of c and e. To take the best case scenario for the parametric model, assume that the measurement error ce has mean zero and is independent of V , W , and e.
Substituting equation (2.7.4) into equation (2.7.3) gives
ln Y = 0+ 1c + 2c2+ W + e (2.7.5)
where
e = 1ce 2 2cce 2c2e+ e.
The error e does not have mean zero and correlates with c and c2, which makes the control function estimator inconsistent. Unlike the case of linear models with independent mean zero measurement errors, the control function estimator is not consistent because of the nonlinearity in this model.
Now consider applying our nonparametric estimator to this model. The treatment indicator D that we would construct is de…ned as equaling one for …rms in the .25 to .75 quantile of c and zero otherwise, while the corresponding indicator D based on the true measure of competitiveness equals one for …rms in the .25 to .75 quantile of c and zero otherwise. Unless the measurement error ceis extremely large, for the large majority of …rms D will equal D . This is part of what makes our estimator more robust to measurement error. Even if all …rms have c mismeasured to some extent, most will still be correctly classi…ed in terms of D.
To check the relative robustness of these estimators to measurement error, we perform additional Monte Carlo analysis. As before, we construct simulated data to match moments
and the sample size of the empirical data set, and to make what would be the true treatment e¤ect in the model match our empirical estimate of 3.9. We do two simulations, one using normal errors and one based on uniform errors, as before. In both, V and W are scaled to have equal magnitudes, so V = 0+ 1"1 and W = 0 + 1"2. To match data moments, the normal error simulations set 0 = 0:375, 1 = 0:0733, and ce = 1"3 where "1, "2, and "3 are independent standard normals and 1 is a constant with values that we vary to obtain di¤erent magnitudes of measurement error. The uniform error simulations set
0 = 1 = 0:25, and ce 2("3 0:5), where now "1, "2, and "3 are independent random variables that are uniformly distributed on [0; 1].
To check for robustness against an alternative speci…cation as well as measurement error, we also generate data replacing the quadratic form in equation (2.7.1) with the step function
ln Y = 0+ ( 1 0)D +ee; (2.7.6)
where D , D, c , c, V , W , and e are all de…ned as above.
The Monte Carlo results, based on 10,000 replications, are reported in Tables 5 and 6 in the supplemental Appendix. In addition to trying out the four estimators we considered earlier, (Trim-ATE, No-Trim-ATE, Naive-ATE, and ML-ATE) we also apply the control function estimator described above, analogous to Hashmi’s estimator.
Our main result is that, with both normal and uniform errors, the greater the magnitude of measurement error is (that is, the larger the 1 and 2 are), the better our estimator performs relative to other estimators. For the quadratic model without measurement error the control function would be a consistent parametric estimator and so should outperforms our semiparametric estimator. We …nd this also holds with very small measurement error
(e.g., 1 = :02 in the left side block of Table 5), however, both control function and Trim-ATE perform about equally at 1 = :03, and at the still modest measurement error level of 2 = :04, Trim-ATE has smaller RMSE (root mean squared error) than all the other estimators, including control function. Similar results hold for the uniform error model reported in Table 6. Also, in the step function model (shown on the right side of Tables 5 and 6) our Trim-ATE is very close to, or superior to, all the other estimators including control functions at all measurement error levels.
It is worth noting that possible measurement error a¤ects our empirical application only because we de…ned treatment D in terms of an observed, possibly mismeasured underlying variable, competitiveness. In other applications the treatment indicator may be observed without error even when an underlying latent measure is completely unobserved. For exam-ple, suppose an outcome Y is determined in part by an individual’s chosen education level, which in turn is determined by an ordered choice speci…cation. The true education level of a student might be unobserved, but a treatment D de…ned as having graduated high school but not college could still be correctly measured.